Matching between Collinear Twist-3 and TMD Fragmentation Function Contributions to Polarized Hyperon Production in SIDIS

TL;DR

The work addresses how collinear twist-3 fragmentation functions and TMD factorization describe the same transverse hyperon polarization effect in SIDIS, specifically for $ep\\to e\\Lambda^\\uparrow X$. It develops the LO twist-3 FF contributions (quark and gluon) and the TMD polarizing FF $D_{1T}^{\perp}$, then computes the perturbative large-$k_\perp$ behavior to express the TMD FF in terms of twist-3 FFs. By matching the small-$P_T$ limit of the twist-3 cross section with the TMD factorization result, the paper demonstrates a consistent, unified description in the intermediate region $\\Lambda_{\\rm QCD} \\ll P_T \\ll Q$, including purely gluonic channels. This finding solidifies the connection between two major QCD frameworks and provides a robust basis for analyzing polarized hyperon production across kinematic regimes.

Abstract

We investigate the consistency between the collinear twist-3 factorization and the transverse-momentum-dependent (TMD) factorization for the transversely polarized hyperon production in semi-inclusive deep inelastic scattering, $ep\to eΛ^\uparrow X$. In particular, we focus on the contributions from the twist-3 fragmentation functions (FFs) and the TMD polarizing FF in the region of the hyperon's intermediate transverse momentum $P_T$, $Λ_{\rm QCD}\ll P_T\ll Q$, where both frameworks are valid. In this region the polarizing FF can be expressed in terms of the twist-3 FFs including the purely gluonic ones, and the resulting TMD factorization formula for $ep\to eΛ^\uparrow X$ agrees with the small-$P_T$ limit of the corresponding twist-3 cross section. This matching of the two calculations indicates that the two frameworks describe the same effect in QCD and provide complementary frameworks for the process in different kinematic regions.

Figures (4)

• Figure 1: Hadron frame
• Figure 2: Feynman diagrams contributing to $D_{1T}^\perp$ at $q_T \gg \Lambda_{\mathrm{QCD}}$. Mirror diagrams also contribute. From left to right, each diagram corresponds to $\Delta (\ell) S(\ell)$, $\Delta_{La}^\alpha (\ell,\ell')S_\alpha^{La}(\ell,\ell')$, $\widehat{\Gamma}_{ab}^{\mu\nu}(\ell) S_{\mu\nu}^{ab}(\ell)$, $\widehat{\Gamma}_{Labc}^{\mu\nu\lambda}(\ell,\ell') S^{Labc}_{\mu\nu\lambda}(\ell,\ell')$, $\widetilde{\Delta}_{La}^\alpha (\ell,\ell') \widetilde{S}^{La}_{\alpha}(\ell,\ell')$. Upper portion in each diagram represents the fragmentation matrix elements (for the twist-3 FFs), while the lower white circles correspond to the partonic hard part. In the second, fourth and fifth diagram, momenta of the parton lines in the left of the cut are $\ell$ and $\ell'-\ell$ flowing upwards.
• Figure 3: Twist-3 quark FF contribution to the TMD polarizing FF at large $k_\perp$. Diagrams in which a gluon line directly connects the twist-3 FF and the Wilson line become zero, so that they are not shown.
• Figure 4: Feynman diagrams for the twist-3 gluon FF contribution to the polarizing FF at large $P_T$. Diagrams in which a gluon line coming out of the upper blob directly couples to the Wilson line become either zero or suppressed at large $\hat{\zeta}^2$, so that they are not shown here.